Introduction to Algebraic Varieties

Course Content
Systems of polynomial equations in many variables define algebraic varieties, which are the basic objects of study in algebraic geometry. In this course we introduce some fundamental notions of algebraic geometry, such as coordinate rings, local rings, function fields, and affine and projective algebraic varieties. We develop the necessary algebraic tools and we prove Hilbert’s Nullstellensatz, the core theorem that links the algebraic and the topological side of algebraic varieties.

The course focuses on the study of one-dimensional algebraic varieties, that is, algebraic curves. Plane curves will be discussed in detail as main concrete examples. For them we prove a theorem of Bézout that answers the question: How many points lie in the intersection of two plane curves? We study morphisms and rational maps between arbitrary curves, and singularities (these occur for example when two branches of the same curve intersect). The goal of the course is to prove the birational classification of algebraic curves via the correspondence between smooth projective curves and function fields of transcendence degree one.

Teaching Methods
Two times 2 hours of lectures per week and two times 2 hours of exercise class per week.

Assessment
The final grade is the maximum of the exam grade and (80% exam + 20% hand-in exercises).

Resit and Effort Requirement
Handing in serious answers to the hand-in assignments and a serious attempt at the exam give the right to do the retake. The final grade after retake is computed the same way; the hand-in counts for the retake as well.

Language
English if desired. If possible, the lectures may be given in Dutch.

Linear algebra (WISB107 en WISB108) and Introduction to groups and rings (WISB124). Recommended: Fields and Galoistheory (WISB224) and Modules and Representations (WISB223). See the courseplanner (cursusplanner.uu.nl) for the contents of those courses: select Faculty of Science and then the programme of the bachelor Mathematics of the most recent year.

• Book Algebraic Curves. William Fulton.